Which expression is equivalent to $\frac{b^{-2}}{b^{-3}}$? Assume $a
eq 0, b
eq 0$.
We are asked to simplify the expression \(\frac{b^{-2}}{b^{-3}}\), given that \(b eq 0\). The expression involves negative exponents, which we can handle using the laws of exponents. Specifically, the quotient rule for exponents states that for any non-zero base \(b\) and exponents \(m\) and \(n\), the quotient \(\frac{b^m}{b^n} = b^{m-n}\). Also, the rule for negative exponents states that \(b^{-n} = \frac{1}{b^n}\).
To simplify \(\frac{b^{-2}}{b^{-3}}\), we apply the quotient rule for exponents directly:
\[\frac{b^{-2}}{b^{-3}} = b^{-2 - (-3)}\]
Next, we perform the arithmetic in the exponent:
\[-2 - (-3) = -2 + 3 = 1\]
Thus, the expression simplifies as follows:
\[\frac{b^{-2}}{b^{-3}} = b^1 = b\]
Alternatively, we can express the terms using positive exponents first:
\[\frac{b^{-2}}{b^{-3}} = \frac{\frac{1}{b^2}}{\frac{1}{b^3}} = \frac{1}{b^2} \cdot \frac{b^3}{1} = \frac{b^3}{b^2} = b\]
This confirms our previous result.
Comparing our result \(b\) to the given options:
None of the provided options are identical to \(b\). If we assume the expression intended was \(\frac{a^3 b^{-2}}{b^{-3}}\), then it would simplify to \(a^3 b\), which matches option (b). However, based strictly on the expression \(\frac{b^{-2}}{b^{-3}}\), the equivalent value is simply \(b\). In the absence of a matching choice, the simplified form is \(\boxed{b}\).
In engineering and computer science, dealing with exponents is crucial for normalizing data or scaling physical quantities. For instance, when calculating the magnitude of a signal or the efficiency of a processor at different energy levels, one might encounter fractions of power functions like \(\frac{P^{-2}}{P^{-3}}\). Simplifying these expressions, as we did here, allows engineers to efficiently determine the base characteristics of the system, simplifying complex calculations down to their fundamental variables.